266 XIV INTEGRATION IN LOCALLY COMPACT GROUPS 4. HAAR MEASURE ON A QUOTIENT GROUP Let G be a locally compact group, G' a closed normal subgroup of G, and G" = G/G' the quotient group, which is locally compact ((12.11.3) and (12.10.9)). Let a', a" be left Haar measures on G', G", respectively, and let 7i ; G -> G" be the canonical homomorphism. (14.4.1) For eachfE Jf (G) and each x e G, the function = ! JG' w continuous on G, tf/7^ #(*<!;) = #(A;) /or a// £ 6 G', .yo that we may write g(x) ~h(n(x)), where h is continuous on G" (12.10.6). The support of h is compact. The positive linear form h(x")da"(x") G" is a left Haar measure on G. The continuity of g follows from (14.1.5.5), and we have because a' is left-invariant. If S = Supp(/), then h(x") ^ 0 implies that x" e 7r(S), and ;r(S) is compact in G" (3.17.9). Finally, if s is any element of G, put/! = yCs"1)/, and \Qtg1,hl be the corresponding functions. Then it is clear that g^(x) = g(sx), and therefore h^x") = h(n(s)xfr). Hence the last part of the proposition follows from the left-invariance of a" (the fact that the linear form /»-*• J ^(^") du."(x") is not zero follows from the fact that the supports of a' and a" are equal to G' and G", respectively). If we denote by a the left Haar measure on G defined by (14.4.1), then by abuse of notation we write (14.4.2) f f(x) <fa(x) - f d<f(x) \ f(x Jo JG" Jo' t) where x = TT(JC), and/e This notational analogy with the product measure (13.21.2) is continued in the following propositions : of Section 8.5). The number a(A) = a+(A) = a~(A)